Burr distribution

In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution and is one of a number of different distributions sometimes called the "generalized log-logistic distribution".

The Burr (Type XII) distribution has probability density function:

${\displaystyle {\begin{aligned}f(x;c,k)&=ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\\[6pt]f(x;c,k,\lambda )&={\frac {ck}{\lambda }}\left({\frac {x}{\lambda }}\right)^{c-1}\left[1+\left({\frac {x}{\lambda }}\right)^{c}\right]^{-k-1}\end{aligned}}}$

The ${\displaystyle \lambda }$parameter scales the underlying variate and is a positive real.

The cumulative distribution function is:

${\displaystyle F(x;c,k)=1-\left(1+x^{c}\right)^{-k}}$

${\displaystyle F(x;c,k,\lambda )=1-\left[1+\left({\frac {x}{\lambda }}\right)^{c}\right]^{-k}}$

It is most commonly used to model household income, see for example: Household income in the U.S. and compare to magenta graph at right.

Given a random variable ${\displaystyle U}$drawn from the uniform distribution in the interval ${\displaystyle \left(0,1\right)}$, the random variable

${\displaystyle X=\lambda \left({\frac {1}{\sqrt[{k}]{1-U}}}-1\right)^{1/c}}$

has a Burr Type XII distribution with parameters ${\displaystyle c}$, ${\displaystyle k}$and ${\displaystyle \lambda }$. This follows from the inverse cumulative distribution function given above.

• When c = 1, the Burr distribution becomes the Lomax distribution.

• When k = 1, the Burr distribution is a log-logistic distribution sometimes referred to as the Fisk distribution, a special case of the Champernowne distribution.

• The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.

• The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution

• Rodriguez, R. N. (1977). "A guide to Burr Type XII distributions". Biometrika. 64 (1): 129–134. doi:10.1093/biomet/64.1.129.

• John (2023-02-16). "The other Burr distributions". www.johndcook.com.